# A nonlinear spectrum on closed manifolds

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Abstract: The p-widths of a closed Riemannian manifold are a nonlinear

analogue of the spectrum of its Laplace--Beltrami operator, which was

defined by Gromov in the 1980s and corresponds to areas of a certain

min-max sequence of hypersurfaces. By a recent theorem of

Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like

the eigenvalues do. However, even though eigenvalues are explicitly

computable for many manifolds, there had previously not been any >=

2-dimensional manifold for which all the p-widths are known. In recent

joint work with Otis Chodosh, we found all p-widths on the round

2-sphere and thus the previously unknown Liokumovich--Marques--Neves

Weyl law constant in dimension 2.